DLMF:18.28.E13 (Q5988)

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DLMF:18.28.E13
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    Statements

    Defining formula
    C n ⁡ ( cos ⁡ θ ; β | q ) = ∑ ℓ = 0 n ( β ; q ) ℓ ⁢ ( β ; q ) n - ℓ ( q ; q ) ℓ ⁢ ( q ; q ) n - ℓ ⁢ e i ⁢ ( n - 2 ⁢ ℓ ) ⁢ θ = ( β ; q ) n ( q ; q ) n ⁢ e i ⁢ n ⁢ θ ⁢ ϕ 1 2 ⁡ ( q - n , β β - 1 ⁢ q 1 - n ; q , β - 1 ⁢ q ⁢ e - 2 ⁢ i ⁢ θ ) . q-ultraspherical-polynomial 𝑛 𝜃 𝛽 𝑞 superscript subscript ℓ 0 𝑛 q-Pochhammer-symbol 𝛽 𝑞 ℓ q-Pochhammer-symbol 𝛽 𝑞 𝑛 ℓ q-Pochhammer-symbol 𝑞 𝑞 ℓ q-Pochhammer-symbol 𝑞 𝑞 𝑛 ℓ superscript 𝑒 imaginary-unit 𝑛 2 ℓ 𝜃 q-Pochhammer-symbol 𝛽 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑒 imaginary-unit 𝑛 𝜃 q-hypergeometric-rphis 2 1 superscript 𝑞 𝑛 𝛽 superscript 𝛽 1 superscript 𝑞 1 𝑛 𝑞 superscript 𝛽 1 𝑞 superscript 𝑒 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle C_{n}\left(\cos\theta;\beta\,|\,q\right)=\sum_{% \ell=0}^{n}\frac{\left(\beta;q\right)_{\ell}\left(\beta;q\right)_{n-\ell}}{% \left(q;q\right)_{\ell}\left(q;q\right)_{n-\ell}}e^{\mathrm{i}(n-2\ell)\theta}% =\frac{\left(\beta;q\right)_{n}}{\left(q;q\right)_{n}}e^{\mathrm{i}n\theta}{{}% _{2}\phi_{1}}\left({q^{-n},\beta\atop\beta^{-1}q^{1-n}};q,\beta^{-1}qe^{-2% \mathrm{i}\theta}\right).}}
    0 references
    DLMF id
    DLMF:18.28.E13
    0 references
    Symbols used
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2addec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
    0 references
    Q11298
    Defining formula
    ( a ; q ) n q-Pochhammer-symbol 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}
    xml-id
    C17.S2.SS1.p1.m2ahdec
    0 references
    basic hypergeometric (or $$q$$ -hypergeometric) function
    Defining formula
    ϕ s r + 1 ⁡ ( a 0 , … , a r ; b 1 , … , b s ; q , z ) q-hypergeometric-rphis 𝑟 1 𝑠 subscript 𝑎 0 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},% \dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}
    xml-id
    C17.S4.E1.m2acdec
    0 references
    Q11728
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C18.S1.XMD3.m1jdec
    0 references
    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1cdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1hdec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1edec
    0 references
     
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